Relative adjoint functors

Idea

Relative adjoints with respect to a functor JJ are a generalization of adjoints, where JJ in the relative case plays the role of the identity in the standard setting: adjoints are the same as IdId-relative adjoints.

Definition

hom-isomorphism definition

Fix a functor J:B→DJ\colon B \to D. Then, a functor

(1)R:C→D
R \colon C \to D

has a left JJ-relative adjoint (or JJ-left adjoint) if there is a functor

Notation

LJ⊣RL {\,\,}_J\!\dashv R stands for LL being the JJ-left adjoint of RR

L⊣JRL \dashv_J R stands for RR being the JJ-right adjoint of LL

absolute lifting definition

Just as with regular adjoints, relative adjoints can be defined in a more conceptual way in terms of absolute liftings. We have

LJ⊣RL {\,\,}_J\!\dashv R if L=LiftRJL = \mathop{Lift}_R J, and this left lifting is absolute

L⊣JRL \dashv_J R if R=RiftLJR = \mathop{Rift}_L J, and this right lifting is absolute

Properties

asymmetry

The most important difference with regular adjunctions is the asymmetry of the concept. First, for LJ⊣RL {\,\,}_J\!\dashv R it makes no sense to ask for R⊣JLR \dashv_J L (domains and comodomains do not typecheck). And secondly, and more importantly:

LL is JJ-left adjoint to RR:RRdeterminesLL

RR is JJ-right adjoint to LL:LLdeterminesRR

(this is obvious from the definition in terms of liftings). Because of this, even if most of the properties of adjunctions have a generalization to the relative setting, they do that in a one-sided way.

Examples

it may happen that HomD(L(−),d)Hom_D(L(-),d) is representable only for somedd, but not for all of them. In that case, taking

(8)J:B→D
J \colon B \to D

be the inclusion of the full subcategory determined by HomD(L(−),d)Hom_D(L(-),d) representable, and defining R:B→CR \colon B \to C accordingly, we have

(9)L⊣JR
L \dashv_J R

This can be specialized to situations such as a category having some but not all limits of some kind, partially defined Kan extensions, etc. See also free object.

fully faithful functors

A functor F:A→BF: A \to B is fully faithful iff it is representably fully faithful iff 1A=LiftFF1_A = \mathop{Lift}_F F, and this lifting is absolute. Thus, FF fully faithful can be expressed as

(10)1F⊣F
1 {\,\,}_F\!\dashv F

nerves

Take AA a locally small category, and F:A→BF\colon A \to B a locally left-small functor (one for which B(Fa,b)B(Fa,b) is always small). The AA-nerve induced by FF is the functor

(11)NF:B→SetAop
N_F \colon B \to \mathbf{Set}^{A^{\mathop{op}}}

given by NF(b)(a)=A(Fa,b)N_F(b)(a) = A(Fa,b). It is a fundamental fact that F=LiftNFyAF = \mathop{Lift}_{N_F} y_A and this lifting is absolute; or, in relative adjoint notation, FyA⊣NFF {\,\,}_{y_A}\!\dashv N_F. The universal 2-cell ι:yA→NFF\iota\colon y_A \to N_F F is given by the action of FF on morphisms:

(12)ιa:yAa→(NFF)(a)
\iota_a \colon y_A a \to (N_F F)(a)

at a′:Aa' \colon A is

(13)Fa,a′:A(a,a′)→B(Fa,Fa′)
F_{a,a'}\colon A(a,a') \to B(Fa, Fa')

Note that when specialized to F=1AF = 1_A, this reduces to the Yoneda lemma: first N1A≃yAN_{1_A} \simeq y_A, and then 1A=LiftyAyA1_A = \mathop{Lift}_{y_A} y_A absolute in hom-isomorphism terms reads:

One of the axioms of a Yoneda structure? on a 2-category abstract over this situation, by requiring the existence of FF-nerves with respect to yoneda embeddings such that the 1-cell FF is an absolute left lifting as above; see Mark Weber or the original Street-Walters papers cited in the references below.